In mathematical logic

Mathematical logic

Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

, Lindström's theorem (named after Swedish logician Per Lindström) states that first-order logic

First-order logic

First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...

 is the strongest logic
(satisfying certain conditions, e.g. closure

Closure (mathematics)

In mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a unique member of the same set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 8 are both natural numbers, but...

 under classical negation

Negation

In logic and mathematics, negation, also called logical complement, is an operation on propositions, truth values, or semantic values more generally. Intuitively, the negation of a proposition is true when that proposition is false, and vice versa. In classical logic negation is normally identified...

) having both the (countable) compactness property

Compactness theorem

In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model...

 and the (downward) Löwenheim–Skolem property

Löwenheim–Skolem theorem

In mathematical logic, the Löwenheim–Skolem theorem, named for Leopold Löwenheim and Thoralf Skolem, states that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ...

.

Lindström's theorem is perhaps the best known result of what later became known as abstract model theory

Abstract model theory

In mathematical logic, abstract model theory is a generalization of model theory which studies the general properties of extensions of first-order logic and their models....

, the basic notion of which is an abstract logic

Abstract logic

In mathematical logic, an abstract logic is a formal system consisting of a class of sentences and a satisfaction relation with specific properties related to occurrence, expansion, isomorphism, renaming and quantification....

; the more general notion of an institution was later introduced, which advances from a set-theoretical notion of model to a category

Category theory

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

theoretical one.